On the existence of canonical multi-phase Brakke flows

Abstract

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established by Fischer et al. in arXiv:2003.05478, under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

48 pages. Comments are very welcome! v5 is the final version, appeared on Adv. Calc. Var

References in corpus (1)

• A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

Cited by in corpus (4)

• Existence of weak solution to volume preserving mean curvature flow in higher dimensions
• Weak-strong uniqueness for the Navier-Stokes equation for two fluids with ninety degree contact angle and same viscosities
• End-time regularity theorem for Brakke flows
• Łojasiewicz-Simon inequalities for minimal networks: stability and convergence